Question

The product of three increasing numbers in

GP

is

5832

. If we add

6

to the second number and

9

to the third number, then resulting numbers form an

AP

. Find the numbers in

GP

Accepted Answer

Let the geometric progression terms be $\frac{a}{r}, a, a r$

Given product of geometric progression terms $= 5832$

$\Rightarrow \frac{a}{r} \cdot a \cdot a r = 5832$

$a^{3} = 5832$

$a^{3} = 2^{3} \times 9^{3}$

$a = 2 \times 9 = 18$

Given $a_{1}, a_{2} + 6, a_{3} + 9$ is in $A P$

$\frac{a}{r}, a + 6, a r + 9$

But $a = 18$

$\frac{18}{r}, 18 + 6, 18 r + 9$

If $A P$ $a_{2} - a_{1} = a_{3} - a_{2}$

$(18 + 6) - \frac{18}{r} = 18 r + 9 - (18 + 6)$

$24 - \frac{18}{r} = 18 r + 9 - 24$

$\frac{24 r - 18}{r} = 18 r - 15$

$24 r - 18 = r (18 r - 15)$

$18 r^{2} - 15 r - 24 r + 18 = 0$

$18 r^{2} - 39 r + 18 = 0$

Divisible by $3$

$6 r^{2} - 13 r + 6 = 0$

$6 r^{2} - 9 r - 4 r + 6 = 0$

$3 r (2 r - 3) - 2 (2 r - 3) = 0$

$(2 r - 3) (3 r - 2) = 0$

$2 r - 3 = 0; 3 r - 2 = 0$

$2 r = 3 \Rightarrow r = \frac{3}{2}; 3 r = 2 \Rightarrow r = \frac{2}{3}$

Terms of the $G P \frac{a}{r}, a, a r$

$a = 18, r = \frac{3}{2}; a = 18, r = \frac{2}{3}$

$a r = 18 \times \frac{3}{2} = 27; \frac{a}{r} = 18 \times \frac{3}{2} = 27$

$a = 18; a = 18$

$\frac{a}{r} = 18 \times \frac{2}{3} = 12; a r = 18 \times \frac{2}{3} = 12$

$12, 18, 27; 27, 18, 12$

Given numbers are in increasing order

Numbers in $G P$ is $12, 18, 27$

Class	$33 - 36$	$37 - 40$	$41 - 44$	$45 - 48$	$49 - 52$
Freq.	$15$	$17$	$21$	$22$	$25$

The product of three increasing numbers in $GP$ is $5832$ . If we add $6$ to the second number and $9$ to the third number, then resulting numbers form an $AP$ . Find the numbers in $GP$

Important Questions on Solved Mathematics Paper-1

Learn and Prepare for any exam you want !

The product of three increasing numbers in GP is 5832. If we add 6 to the second number and 9 to the third number, then resulting numbers form an AP. Find the numbers in GP

Important Questions on Solved Mathematics Paper-1

Learn and Prepare for any exam you want !

The product of three increasing numbers in $GP$ is $5832$ . If we add $6$ to the second number and $9$ to the third number, then resulting numbers form an $AP$ . Find the numbers in $GP$